Nonsteady interior ballistics of cylindrical-grain solid rocket motors

Transcription

1 Computational Ballistics II 281 Nonsteady interior allistics of cylindrical-grain solid rocket motors D. R. Greatrix Department of Aerospace Engineering, Ryerson University, Canada Astract A numerical model for the prediction of undesirale axial comustion instaility symptoms in solid-propellant rocket motors is presented. The unsteady interior flow in the motor chamer, the oscillation of the surrounding structure of the propellant and motor casing, and the corresponding transient comustion process, are all incorporated within this model. The dynamic comustion process is modelled via the Zeldovich-Novozhilov solid-phase energy conservation approach, with a urning rate limiting function that allows for an applicale alignment with oserved experimental comustion response characteristics of a given propellant. Example results from the overall numerical model are presented for cylindrical-grain motors having differing characteristics with respect to geometry and urning rate. Keywords: solid rocket motor, comustion instaility, transient flow. 1 Introduction Research towards predicting and quantifying undesirale axial comustion instaility symptoms in solid-propellant rocket motors (SRMs) necessitates comprehensive numerical models for interior allistic simulation under dynamic flow and comustion conditions. An effective model comines the effects of the unsteady flow, the transient comustion process, and the structural dynamics of the surrounding propellant/casing structure. In the present investigation, a numerical model incorporating the aove attriutes is used in the prediction of the unsteady instaility-related ehaviour in cylindrical-grain motors. Pertinent key parameters, like the urning rate limiting coefficient set the in the Zeldovich- Novozhilov (Z-N) transient urning model (De Luca et al [1], Yang et al [2], Greatrix [3]) are varied in a series of test simulations, and evaluated for any

2 282 Computational Ballistics II identifiale trend as relates to the prediction of axial comustion instaility symptoms in actual motors. Figure 1: SRM model setup. 2 Method A simplified schematic diagram of the physical system of an SRM placed on a static test stand is provided in fig. 1. In this example, the cylindrical-grain motor is free to virate radially without any external constraint (i.e., only constrained as indicated y a thick steel static-test sleeve surrounding the aluminum flightweight casing), while axial motion is constrained to a large degree y the thrust-measuring load cell (represented here as a spring/damper) at the lefthand oundary. Under nominal operating conditions, the internal gas flow moves smoothly from the urning propellant surface through and eyond the exhaust nozzle. With respect to modelling the internal flow within the motor, a numer of studies for SRMs having larger length-to-diameter (L/D) ratios have incorporated the use of a one-dimensional unsteady flow model, e.g., as done y Loncaric et al [4] using a higher-order random-choice method (RCM) for the flow solution. The effect of such factors as turulence can e included through one or more additional equations that employ the information from the ulk flow properties arising from the one-dimensional equations of motion for the gas (and for the particles, if a two-phase flow solution). In those cases where two- or three-dimensional flow effects can influence a particular aspect of the motor s interior allistic ehaviour, one might consider a multi-dimensional flow solution approach if a one-dimensional solution does not capture (or capture adequately) the phenomenon in question. For the purposes of the present study, a one-dimensional RCM approach will e utilized for the unsteady flow solution. Structural viration can play a significant role in nonsteady SRM interior allistic ehaviour, as evidenced y oserved changes in comustion instaility symptoms as allied to changes in the structure surrounding the internal flow (e.g., propellant grain configuration, wall thickness, material properties). The level of sophistication required for modelling the motor structure (propellant, casing, static-test sleeve, nozzle) and applicale oundary conditions (load cell on static test stand) can vary, depending on the particular application and motor

3 Computational Ballistics II 283 design. Loncaric et al [4] employed a finite-element approach towards the structural modelling of a star-grain propellant configuration. In the present study, a cylindrical-grain configuration allows for a simpler approach from thickwall theory, as reported y Greatrix and Harris [5]. The radial deformation dynamics of the propellant/casing/sleeve are modelled y a series of independent ring elements along the length of the motor. Axial motion along the length of the structure is modelled via eam theory, and ounded y the spring/damper load cell at the motor s head end. Viscous damping is applied in the radial and axial directions. Reference structural properties are assumed for an ammoniumperchlorate/hydroxyl-terminated polyutadiene (AP/HTPB) composite propellant surrounded y an aluminum casing and steel sleeve. For greater accuracy, some properties like the propellant/casing/sleeve assemly s natural radial frequency may e predetermined via a finite-element numerical solution, rather than via theoretical approximations. The transient urning rate for the regression of the core periphery of the cylindrical propellant grain may e modelled through the Z-N approach. This phenomenological modelling technique applies an energy conservation criterion in coupling the heat conduction within the solid phase (the propellant) to the heat produced in the gas phase aove the propellant surface. Conveniently, empirical or semi-empirical steady-state urning rate information may e used in place of more complex dynamic flame-ased reaction rate equations. The principal equation for the nominal (or unconstrained) instantaneous urning rate r *, tying the solid phase to the gas phase, is given y: 0 1 r = r,qs Tdx (1) (T T ) t s where r,qs is the quasi-steady urning rate (value for urning rate as estimated from steady-state information for a given set of local flow conditions), T s is the urning surface temperature at the spatial position x = 0, T i is the initial propellant temperature (x< 0 moving deeper into the propellant), and )T = T(x,t) T i is the temperature distriution within the propellant. The transient heat conduction in the solid phase can e solved y an appropriate finite-difference scheme. As reported y Greatrix [3], the actual instantaneous urning rate r may e found as a function of r * through the rate limiting equation: i dr dt = K ( r r ) (2) The rate limiting coefficient K effectively damps the unconstrained urning rate r * when for a finite time increment t : K 1 < (3) t

4 284 Computational Ballistics II The quasi-steady urning rate r,qs can e found as a function of various parameters; in this study, as a function of local static pressure p, core flow velocity u (erosive urning component), and normal/lateral/longitudinal acceleration such that: r = r + r + r (4),qs p e a The equations needed for the coupled solution of r,qs may e found in [4]. 3 Results and discussion The reference SRM for this investigation is a cylindrical-grain motor employing a nonaluminized AP/HTPB propellant, with characteristics as reported y Greatrix and Harris [5]; some parameters have een updated or added for the current effort. A thin steel sleeve of 4.7-mm thickness is in place for the reference static-test simulations. In an actual test firing for evaluation of an SRM s susceptiility to nonlinear axial comustion instaility symptoms, after the initial pulse disturance is introduced into the flow of the motor chamer, an unstale motor will exhiit a sustained axial compression wave, on occasion accompanied y a sustantial increase in the ase chamer pressure (referred to as a dc shift). An initial pulsed-firing simulation run was completed as a starting point for this study, in which no transient lag/lead effects on comustion were included, i.e., local urning rate r = r,qs. In fig. 2 for head-end pressure p c as a function of time somewhat later into the firing such that the principal shock has reached its quasi-equilirium strength, one can see the sustained shock wave front arriving aout every 1 ms, oscillating at the fundamental axial resonant frequency f 1L of 1 khz. The ase pressure is sustantially elevated over the nominal operating chamer pressure. The decay in pressure after each shock front arrival is quite rapid in this example, susiding to the ase pressure level well efore the next shock wave front arrival. The shock wave in this case is sustained primarily due to the heightened urning rate augmentation resulting from the radial viration of the urning propellant surface. Moving to the case where the transient solid/gas-phase response of the reference solid propellant is incorporated in the numerical solution, one can refer to [3] for some ackground results for frequency response trends as relates to the rate limiting coefficient K and ase urning rate. Below a certain threshold value for K, the transient response of the propellant is too sluggish to sustain an axial pressure wave. For the particular reference motor and pulsing conditions under evaluation here, a value for K of s -1 appears to e close to the threshold needed for the axial wave to persist. The leading wave front arising from the initial travelling pulse introduction into the flow struggles to reach a quasi-equilirium level, and then decays. At a value for K of s -1, the motor is capale of sustaining a strong axial pressure wave moving at the fundamental axial resonant frequency f 1L of 1 khz. As illustrated y the headend pressure-time profile later into the firing of fig. 3, the shock front is followed y a more gradual susidence in pressure when compared to fig. 2; the results of

5 Computational Ballistics II 285 fig. 3 might e said to e more comparale to experimental oservation in this respect. In this case, it is the transient response of the propellant that slows this cyclic drop in pressure. That eing said, the heightened response of the propellant s urning rate to dynamic normal acceleration is still necessary for the wave to e sustained; in the asence of any acceleration-related urning, the wave after initial pulse introduction would continue to decay. 14 p c (MPa) t (s) Figure 2: Head-end pressure-time profile, no transient Z-N response. At a value for K of s -1, the motor s instaility symptoms transform towards a preference to a two-shock system, versus the one-shock system aove. While initially pulsed y a travelling pressure disturance that would firstly favour a single wave development, the transient comustion response mechanism s increase in resonant frequency f r,zn with a higher K, towards a etter coupling with a two-shock flow system (i.e., 2 khz axial resonant frequency f 2L for the gas cavity), in this case overrides the initial pulsing conditions. As shown y the pressure-time profile of fig. 4 later into the firing simulation, one oserves the arrival of a shock front every 0.5 ms at the head end; the ase pressure level is sustantially higher for that time in the firing, versus the previous cases. As efore, one should note that without accelerationrelating urning, the two axial waves would not have een sustained here; with initial mid-motor upstream/downstream pulses introduced to the flow that would initially favour a two-shock flow system, the result would still e wave susidence. As one might expect, increasing values for K do act to slow the

6 286 Computational Ballistics II eventual decay of the axial wave system. In the alternative case, in the asence of a Z-N transient comustion response, the system favours a single wave system for sustained symptoms, possily due to lower wave interference versus a two-wave flow system. As one increases K towards s -1, without acceleration-related urning, a pattern of the wave system devolving and lining up (and decaying) at the resonant frequency of the Z-N comustion mechanism is oserved. 15 p c (MPa) t (s) Figure 3: Predicted pressure-time profile, K = s -1. A second cylindrical-grain motor example involves a shorter motor length (grain length just over 25 cm, vs. 50 cm) and smaller nozzle throat diameter (1.12 cm, vs. 1.6 cm). These dimensions were chosen in order to ring the gas cavity resonant frequency closer to the Z-N comustion mechanism s resonant frequency, for a comparale chamer pressure of just over 10 MPa. Likely due to the increase in the value for f 1L to aout 2 khz from the original 1 khz of the longer motor descried earlier, the shorter motor fails to sustain an axial shock wave at a K of s -1, where the corresponding f r,zn is on the order of 1.1 khz. As illustrated y the results of fig. 5, at a higher K setting of s -1 and a corresponding f r,zn on the order of 1.25 khz, the motor is now capale of sustaining a single-wave system. In the asence of acceleration-related urning for these calculations, it is evident that a much higher value for K is needed for any sustained wave motion in the motor chamer, even when the gas cavity resonant frequencies f 1L and f 2L (single or two-wave system) are more closely aligned to the frequency f r,zn of a given K.

7 Computational Ballistics II p c (MPa) 15 Figure 4: Predicted pressure-time profile, K = s -1. In ending this discussion of results, it should e noted that the transient urning model utilized for this study, from [3], is undergoing further development. For example, in order to render the model independent of time ( t) and spatial ( x) increment sizing in the finite-difference scheme for heat conduction in the solid propellant, the oundary condition for heat input y the gas phase at the propellant surface should e: q = ( K t ) ρ C ( r r )(T T ) (5) eff s s With this particular modification incorporated into the simulation, the setting for the rate limiting coefficient K would increase somewhat for a comparale result, e.g., from s -1 to s Concluding remarks t (s) The implications of such factors as structural viration and frequency-dependent comustion response on nonlinear axial comustion instaility symptom development have een demonstrated y the numerical simulation results presented in this study of cylindrical-grain solid rocket motors. Single and twowave flow systems have appeared in the present results, apparently dependent at least in part on the selected value for the rate limiting coefficient, and the corresponding resonant conditions for the comustion response mechanism and the gas cavity. The results presented in this paper tend to confirm the importance,qs s i

8 288 Computational Ballistics II of acceleration-related urning in sustaining instaility symptoms, with or without the influence of transient frequency-dependent comustion response. Unquestionaly, further work remains to e done in estalishing a more complete understanding of the various mechanisms involved in driving instaility symptoms. More explicit capturing of such influences as radial waves, and heightened acceleration levels of the structure at various locations and times (Greatrix [6]; Chopra et al [7]) through the use of more sophisticated, multi-dimensional numerical models may result in etter and more precise predictions of instaility symptoms. 20 p c (MPa) t (s) Figure 5: Predicted pressure-time profile, K = s -1, short motor. References [1] De Luca, L., Price, E.W. and Summerfield, M. (eds.), Nonsteady urning and comustion staility of solid propellants. Progress in Astronautics & Aeronautics, Vol. 143, AIAA, pp , , , , [2] Yang, V., Brill, T.B. and Ren, W.Z. (eds.), Solid propellant chemistry, comustion, and motor interior allistics. Progress in Astronautics & Aeronautics, Vol. 185, AIAA, pp , , [3] Greatrix, D.R., Numerical transient urning rate model for solid rocket motor simulations. AIAA Paper No , 2004.

9 Computational Ballistics II 289 [4] Loncaric, S., Greatrix, D.R. and Fawaz, Z., Star-grain rocket motor nonsteady internal allistics. Aerospace Science & Technology, 8(2), pp , [5] Greatrix, D.R. and Harris, P.G., Structural viration considerations for solid rocket internal allistics modeling. AIAA Paper No , [6] Greatrix, D.R., Fluid-structure interactions in solid rocket. 13th Propulsion Symposium, 50th CASI Annual Conference, [7] Chopra, H.S., Greatrix, D.R. and Kawall, J.G., Transient shock wave interaction with rocket nozzle cold-flow study. AIAA Paper No , 2003.